Arboles Binarios– Parte 2 - Ciencias Computacionales

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Propedéutico de
Programación
Coordinación de Ciencias
Computacionales
Semana 6,
Primera Parte
Dra. Pilar Gómez Gil
http://ccc.inaoep.mx/~pgomez/cursos/programacion/
Versión 1.2 08.07.08
Chapter 8
Binary Search Trees
Continuación…
Tomado de: Dale, N. Weems, C++ Plus Data Structures
4th. Ed. 2007 Instructor material, © Jones & Barlett Publishers
Tree Recursion
CountNodes Version 1
if (Left(tree) is NULL) AND (Right(tree) is NULL)
return 1
else
return CountNodes(Left(tree)) +
CountNodes(Right(tree)) + 1
What happens when Left(tree) is NULL?
Tomado de: Dale, N. Weems, C++ Plus Data Structures 4th.
Ed. 2007 Instructor material, © Jones & Barlett Publishers
Tree Recursion
return 1
else if Left(tree) is NULL
return CountNodes(Right(tree)) + 1
else if Right(tree) is NULL
return CountNodes(Left(tree)) + 1
else return CountNodes(Left(tree)) +
What happens when the initial tree is NULL?
Tree Recursion
if tree is NULL
return 0
else if (Left(tree) is NULL) AND (Right(tree) is NULL)
return 1
return CountNodes(Right(tree)) + 1
return CountNodes(Left(tree)) + 1
else return CountNodes(Left(tree)) + CountNodes(Right(tree)) +
1
Can we simplify this algorithm?
Tree Recursion
if tree is NULL
return 0
else
return CountNodes(Left(tree)) +
Is that all there is?
// Implementation of Final Version
int CountNodes(TreeNode* tree); // Pototype
int TreeType::LengthIs() const
// Class member function
{
return CountNodes(root);
}
int CountNodes(TreeNode* tree)
// Recursive function that counts the nodes
{
if (tree == NULL)
return 0;
else
return CountNodes(tree->left) +
CountNodes(tree->right) + 1;
Tomado de: Dale, N. Weems, C++ Plus Data Structures 4th. Ed.
}
2007 Instructor material, © Jones & Barlett Publishers
Retrieval Operation
Retrieval Operation
void TreeType::RetrieveItem(ItemType& item, bool& found)
{
Retrieve(root, item, found);
}
void Retrieve(TreeNode* tree,
ItemType& item, bool& found)
{
if (tree == NULL)
found = false;
else if (item < tree->info)
Retrieve(tree->left, item, found);
Retrieval Operation, cont.
else if (item > tree->info)
Retrieve(tree->right, item, found);
else
{
item = tree->info;
found = true;
}
}
The Insert Operation
• A new node is always inserted into
its appropriate position in the tree as
a leaf.
Insertions into a Binary Search Tree
Tomado de: Dale, N. Weems, C++ Plus Data Structures 4th. Ed. 2007 Instructor
material, © Jones & Barlett Publishers
The recursive InsertItem operation
Por ejemplo,
insertando11…
Tomado de: Dale, N. Weems, C++ Plus Data Structures 4th. Ed. 2007
Instructor material, © Jones & Barlett Publishers
The tree parameter
is a pointer within the tree
Recursive Insert
void Insert(TreeNode*& tree, ItemType item)
{
if (tree == NULL)
{// Insertion place found.
tree = new TreeNode;
tree->right = NULL;
tree->left = NULL;
tree->info = item;
}
else if (item < tree->info)
Insert(tree->left, item);
else
Insert(tree->right, item);
}
Eliminando un Nodo
• Hay 3 posibles casos:
1. Que el nodo a eliminarse no tenga
ningún hijo
2. Que el nodo a eliminarse tenga un
hijo
3. Que el nodo a eliminarse tenga dos
hijos
Deleting a Leaf Node
Caso
1
La eliminación es simplemente limpiar
memoria
Deleting a Node with One Child
Caso
2
La eliminación consiste en que el hijo toma el lugar del padre…
Deleting a Node with Two Children
Caso
3
Hay que determinar quien toma el
lugar del padre sin que se afecte el árbol…
Q
Para el caso 3
• El lugar del padre eliminándose, lo puede
tomar su predecesor inmediato o su
sucesor inmediato.
• En éste caso se utiliza al predecesor, el
cual es el valor mas grande en el subárbol
izquierdo del elemento eliminándose, esto
es, el nodo que está mas hacia la derecha
en el subárbol izquierdo
DeleteNode Algorithm
Set tree to NULL
Set tree to Right(tree)
Set tree to Left(tree)
else
Find predecessor
Set Info(tree) to Info(predecessor)
Delete predecessor
Code for DeleteNode
void DeleteNode(TreeNode*& tree)
{
ItemType data;
TreeNode* tempPtr;
tempPtr = tree;
if (tree->left == NULL) {
tree = tree->right;
delete tempPtr; }
else if (tree->right == NULL){
tree = tree->left;
delete tempPtr;}
els{
GetPredecessor(tree->left, data);
tree->info = data;
Delete(tree->left, data);}
Tomado de: Dale, N. Weems,
}
C++ Plus Data Structures
• Pero primero hay que encontrar el
nodo a eliminarse… recordando:
DeleteItem(itemType item)
• Función: Borra el elemento cuya llave coincide con la llave
de item
• Precondición: La llave de item se inicializó. Uno y solo uno
de los elementos en el árbol tiene una llave que coincide
con la de item
• Postcondición: Ningún elemento en el árbol tiene una llave
que coincide con la de item
Definition of Recursive Delete
Definition:
Size:
Removes item from tree
The number of nodes in the path from the
root to the node to be deleted.
Base Case:
If item's key matches key in Info(tree),
delete node pointed to by tree.
General Case:
If item < Info(tree),
Delete(Left(tree), item);
else
Delete(Right(tree), item).
Código
void TreeType::DeleteItem(ItemType
item)
// Calls recursive function Delete to
delete item from tree.
{
Delete(root, item);
}
Tomado de: Dale, N. Weems, C++ Plus Data
Structures 4th. © Jones & Barlett Publishers
Code for Recursive Delete
void Delete(TreeNode*& tree, ItemType item)
{
if (item < tree->info)
Delete(tree->left, item);
else if (item > tree->info)
Delete(tree->right, item);
else
DeleteNode(tree); // Node found
}
Code for GetPredecessor
void GetPredecessor(TreeNode* tree,
ItemType& data)
{
while (tree->right != NULL)
tree = tree->right;
data = tree->info;
}
Printing all the Nodes in
Order
Function Print
Definition:
Prints the items in the binary
search tree in order from
smallest to largest.
Size:
The number of nodes in the
tree whose root is tree
Base Case: If tree = NULL, do nothing.
General Case: Traverse the left subtree in order.
Then print Info(tree).
Then traverse the right subtree in
order.
Code for Recursive InOrder
Print
void PrintTree(TreeNode* tree,
std::ofstream& outFile)
{
if (tree != NULL)
{
PrintTree(tree->left, outFile);
outFile << tree->info;
PrintTree(tree->right, outFile);
}
}
Destructor
void Destroy(TreeNode*& tree);
TreeType::~TreeType()
{
Destroy(root);
}
void Destroy(TreeNode*& tree)
{
if (tree != NULL)
{
Destroy(tree->left);
Destroy(tree->right);
delete tree;
}
}
Algorithm for Copying a
Tree
if (originalTree is NULL)
Set copy to NULL
else
Set Info(copy) to Info(originalTree)
Set Left(copy) to Left(originalTree)
Set Right(copy) to Right(originalTree)
Code for CopyTree
void CopyTree(TreeNode*& copy,
const TreeNode* originalTree)
{
if (originalTree == NULL)
copy = NULL;
else
{
copy = new TreeNode;
copy->info = originalTree->info;
CopyTree(copy->left, originalTree->left);
CopyTree(copy->right, originalTree->right);
}
}
Recorrido recursivo de árboles
binarios
- Preorder
Raíz
Árbol
Izquierdo
Árbol
Derecho
- Inorder
Árbol
Izquierdo
Raíz
Árbol
Derecho
- Postorder
Árbol
Izquierdo
Árbol
Derecho
Raíz
Inorder(tree)
if tree is not NULL
Inorder(Left(tree))
Visit Info(tree)
Inorder(Right(tree))
To print in alphabetical order
Postorder(tree)
if tree is not NULL
Postorder(Left(tree))
Postorder(Right(tree))
Visit Info(tree)
Preorder(tree)
if tree is not NULL
Visit Info(tree)
Preorder(Left(tree))
Preorder(Right(tree))
Three Tree Traversals
Our Iteration Approach
• The client program passes the ResetTree and
GetNextItem functions a parameter indicating
which of the three traversals to use
• ResetTree generates a queues of node contents
in the indicated order
• GetNextItem processes the node contents from
the
appropriate queue: inQue, preQue,
postQue.
Code for ResetTree
void TreeType::ResetTree(OrderType order)
// Calls function to create a queue of the tree
// elements in the desired order.
{
switch (order)
{
case PRE_ORDER : PreOrder(root, preQue);
break;
case IN_ORDER : InOrder(root, inQue);
break;
case POST_ORDER: PostOrder(root, postQue);
break;
}
}
Code for GetNextItem
void TreeType::GetNextItem(ItemType& item,
OrderType order,bool& finished)
{
finished = false;
switch (order)
{
case PRE_ORDER : preQue.Dequeue(item);
if (preQue.IsEmpty())
finished = true;
break;
case IN_ORDER
: inQue.Dequeue(item);
if (inQue.IsEmpty())
finished = true;
break;
case POST_ORDER: postQue.Dequeue(item);
if (postQue.IsEmpty())
finished = true;
break;
}
}
• Ver
– TreeType.h
– TreeType.cpp

Arboles Binarios– Parte 2 - Ciencias Computacionales

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